the tiifinilejlmal Calculus. 34^ 



THjlinBion of Mathematical Infinity into Scnfible andAbfolute. 



41. This observation leads to the difiin&ion of mathema- 

 tical infinity into two kinds, namely, J'enjible, or ajjignablcj 

 infinity, and abfolutc, or mctaphyjical, infinity, which is the 

 limit of the former. 



If then, to any infinitely fmall quantity, be afligned a 

 determinate value, which is not o, this value will be what I 

 call a fenjible, or ajjignablc, Infinitefimal ; whereas, if this 

 value be the laft of all, that is, if it be absolutely nothing, it 

 will be what I call an abfolute, or mctaphyjical, Infinitefimal, 

 which I (hall alfo diftinguifh by the name of an evanefcent 

 quantity. Thus an evanefcent quantity is not that which is 

 generally called an infinitely fmall quantity^ but only the 1 

 ultimate value of that quantity. It is only, I fay, a deter- 

 minate value which, like any other value, may be attributed 

 to that arbitrary quantity, which is generally denominated 

 infinitely fmall. 



42. The confederation of thefe evanefcent quantities would 

 be almoft ufelefs, if in calculation we were refhicted to treat 

 them as fimple nullities ; for, in that cafe, they would pre- 

 fent only the vague ratio of o to o, which is no more equaf 

 to 2 than it is to 3, or to any other quantity whatloever. 

 But it muft not be forgotten, that thefe nullities are here 

 inverted with particular properties, as the ultimate value of 

 indefinitely fmall quantities, whofe limits they are; and that 

 the particular epithet, evanefcent, is applied to them in order 

 to denote, that, of all the ratios and relations of which they 

 are fufceptible in quality of nullities, no other is confidered 

 as entering into the calculation, than thofe which the law of 

 continuity afligns to them, when the fyftem of auxiliary 

 quantities is fuppofed infenfibly to approach to the fyftem of 

 afligned quantities. This idea is what fome great geometri- 

 cians have thought they could exprefs, when they faid, that 

 evanefcent quantities were quantities confidered, neither 

 before nor after they had vanifhed, but in the very inftant of 

 their vanishing *. 



For 



* (C'efl ce que dc grands gcomttres onl cru powwir expiimcr, Sec. ) 

 The author h«r« plainly alludes to Sir I. Ncw;on, the author of this 



doctrine 



